Power Constrained Design Optimization of Analog Circuits Based on Physical gm/ID Characteristics Alessandro Girardi and Sergio Bampi Federal University of Rio Grande do Sul - UFRGS PGMICRO, Informatics Institute Caixa Postal 15.064 - Zip 91.501-970 Porto Alegre-RS, Brazil {girardi,
bampi}@inf.ufrgs.br
ABSTRACT
tives (multiple and complex performance specifications to be met, like bandwidth, gain, power, maximum offset voltage, power supply rejection ratio, etc.) and the large number of design variables (transistor sizes and bias currents) to be set. Previous work has been done in the field of analog design automation to enable fast design at the block level. Different strategies and approaches have been used, such as symbolic simulation [8], artificial intelligence [6], manually derived design equations [5], hierarchy and topology selection [10] and geometric programming [11]. The main difficulty encountered for widespread use of these tools is that they require appropriate modeling of both the devices (technology dependence) and of the circuit in order to achieve the design objectives in a reasonable processing time. The choice of the different circuit topologies to support in a method or tool is also a problem, since most approaches work with topologybased equations, which limits the application range. The addition of new block topologies has to be supported, and requires again expert analog designer knowledge. The use of optimization algorithms combined with physical models seems to be a good solution when applied to specific applications, since a general solution most often proves to have shortcomings for fully exploiting the capabilities of the analog CMOS technology. The main requirements of an analog synthesis tool are: user interactivity, flexibility for multiple topologies, connection to automatic layout generation, and reasonable response time. The interface with an electrical simulator is also convenient. This paper describes a methodology for analog design automation that combines the simulated annealing optimization technique, a physicsbased transconductance-to-current ratio characteristics and the electrical simulation integrated in the LIT layout generation tool [9], providing even to a non-expert user a very flexible tool that is able to size analog circuits, including a broad range of analog constraints, including total power dissipation. The paper is organized as follows: Section 2 describes the proposed methodology and explains how the simulated annealing algorithm, the gm/ID characteristics and the ACM MOSFET model are used to search optimized designs; Section 3 shows the synthesis of a two-stage Miller operational amplifier, in order to demonstrate the capabilities of the methodology; finally, Section 4 summarizes the main contributions.
This paper presents a transistor optimization methodology for low-power analog integrated CMOS circuits, relying on the physics-based gm/ID characteristics as a design optimization guide. Our custom layout tool LIT implements and uses the ACM MOS compact model in the optimization loop. The methodology is implemented for automation within LIT and exploits all design space through the simulated annealing optimization process, providing solutions close to optimum with a single technology-dependent curve and accurate expressions for transconductance and current valid in all operation regions. The compact model itself contributes to convergence and to optimized implementations, since it has analytic expressions which are continuous in all current regimes, including weak and moderate inversion. The advantage of constraining the optimization within a power budget is of great importance for low-power CMOS. As examples we show the optimization results obtained with LIT, resulting in significant power savings, for the design of a two-stage Miller operational amplifier.
Categories and Subject Descriptors B.7.1 [Integrated Circuits]: Types and Design Styles; B.7.2 [Integrated Circuits]: Design Aids—Layout, Simulation
General Terms Performance, Design, Reliability
Keywords simulated annealing, analog design, synthesis
1.
INTRODUCTION
The design of CMOS analog integrated circuits is demanding due to the complex relations between the design objec-
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2.
ANALOG DESIGN METHODOLOGY
The design of analog integrated circuits requires extensive design practice with a given technology to correctly size transistors in order to achieve the required performance. Analytical knowledge-based equations describe the relations between the transistors (design parameters), design specifications (e.g. slew-rate more than 10V /μs) and design objectives (such as minimum power, area, noise, etc, or a combination thereof). These equations are topology-specific and can be used within an automatic synthesis methodology, which must perform the resolution of a system of nonlinear equations. This system usually has more independent variables than equations, returning a wide solution space. The search for an optimum design, however, is often made by extensive simulation practice, by expert heuristics and, to a much lesser extent, by optimization algorithms. The simulated annealing (SA) algorithm was implemented as an option to optimize the design, since it exploits the entire design space, including different transistor lengths. In most devices these are kept fixed by the tools or the designer. SA is a well known random-search technique which exploits an analogy between the way in which a metal cools and freezes into a minimum energy crystalline structure (the annealing process) and the search for a minimum of a cost function in a more general system. It forms the basis of an optimization technique for combinatorial and other problems [12]. The use of simulated annealing in the synthesis of analog circuits was reported in previous works [7]. SA’s major advantage over other methods is the ability to avoid becoming trapped in local minima in the parameter space. The algorithm employs a random search which does not only accept solutions that decrease the objective cost function fc (assuming a minimization problem), but also some changes that increase it. In the design procedure herein proposed, a methodology called gm/ID is used for the circuit performance evaluation. This methodology considers the relationship between the ratio of the transconductance gm over DC drain current ID and the normalized drain current In = ID /(W/L) as a fundamental design parameter [13], such as the measured curve shown in figure 1. The gm/ID characteristic is directly related to the performance of the transistors, gives a clear indication of the device operation region and provides a way for straightforward estimation of transistors dimensions. The main advantage of this method is that the gm/ID x In curve is unique for a given technology, reducing the number of electrical parameters related to the fabrication process. Additionally, its analytical form covers all transistor operation regimes, from weak to moderate to strong inversion. In the LIT tool, the gm/ID x In curve is automatically evaluated by electrical simulation using the ACM compact MOSFET model [3], which is also implemented in the commercial simulator SMASH. An automatic parameter conversion procedure from BSIM3 to ACM is reported [2], warranting the compatibility with most technology kits provided by silicon foundries. The analog circuit modeling for simulated annealing is straightforward. The design objective, or its cost function, has to be formulated appropriately and then minimized. In this work we propose and use the following cost function:
fc =
n X i=1
αi pˆi (X) +
m X
βj cˆj (X)
Figure 1: Measured gm/ID x In curve for NMOS 0.35μm CMOS technology. where αi is the weighting coefficient for performance parameter pˆi (X) , which is a normalized function of the vector of independent design parameters X. This function allows the designer to set the relative importance of competing performance parameters, such as, for example, a weighted relation between power and area. The parameter cˆj (X) is a constraint normalized function, which shrinks the design space to feasible solutions of design specifications. The coefficient βj indicates how closely the specification must be pursued. If cˆj (X) is inside a given specification, it is set to zero. The correct design space exploration is directly related to the cost function formulation [4]. Figure 2 shows the proposed design flow. The user enters the design specifications, technology parameters and configures the cost function according to the required design objectives and specifications. The optimization loop performs a random perturbation on the design variables, whose amplitude is defined by the ”temperature”. These variables are defined by the user, and are always related to the transistor geometry, large and smallsignal parameters, such as W , L, ID , gm and gm/ID . Following, the design properties evaluation is performed by the calculation of the circuit characteristics such as gain, cutoff frequency, phase margin, power, common-mode range, etc. This is done using circuit-specific analytical equations, the gm/ID versus In curve and the ACM model for calculation of transconductances, drain-source saturation voltages and currents. If the circuit is feasible, i.e., transistor sizes are within an allowed range, the cost function can be evaluated and the solution is accepted if the cost decreased or else if the cost increased with a given probability to avoid trapping in a local minima. The final solution returns the devices dimensions. The entire automatic optimization flow is implemented in the LIT tool. The tool returns a spicelike description of the sized circuit and the evaluated performance. External electrical simulations are then evaluated at this stage, in order to verify the solution integrity. The physical synthesis can also be performed in the LIT tool, which generates the layout according to specific criteria of transistor pair matching, transistor folding/splitting, and even more complex associations of transistors.
(1)
j=1
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Figure 3: Schematics of a two-stage operational amplifier Figure 2: Proposed design flow
3.
DESIGN EXAMPLE: MILLER AMPLIFIER
The proposed algorithm and design methodology were implemented and applied to the synthesis of a two-stage operational amplifier, shown in figure 3. This amplifier is composed by an input differential pair with active load in the first stage, an inverter amplifier in the second stage, and a compensation capacitor for stability, connecting nodes 4 and 5 between first and 2nd stages. The analytical equations that describe the behavior of this circuit are well-known [1]. In this example, we want to size the transistors in order to achieve the design specifications given in Table 1. The design objective (fc ) is to minimize the relative area and total DC current, and to maximize the low-frequency gain Av , in the following way:
“W ” L
Av =
gm ID
L W1 =
1
=
(4) 1
ID1 In1
“W ” L
1
· L1
(7)
“W ”
“ gm ” ID
1
· (V A1 + V A3 ) ·
Table 1: Specifications and Two-Stage Amplifier Specification Total current (μA) Phase margin (◦ ) Low-frequency gain (dB) GBW (M Hz) Slew-rate (V /μs) ICMR− (V ) ICMR+ (V ) Total area (μm2 ) Offset (μV ) Cost function
So, the aspect ratio for the input transistors is: “W ”
ID1 In3
“ gm ” ID
5
· (V A5 + V A6 ) (9)
(3)
The drain current for these transistors can be calculated with the information about the transconductance-to-current ratio, which is an independent variable: gm1 ”
=
Our LIT tool implements the ACM model source code and can estimate the Early voltage (V A) according to the transistor length. In this example, the technology used is CMOS 0.35μm, the power supply voltage is ±1.65V and the load capacitance is 10pF . The independent variables subjected to perturbations by the simulated annealing algorithm are: L1 = L2 , L3 = L4 , L5 , L6 , L7 = L8 , (gm/ID )1 = (gm/ID )2 , (gm/ID )3 = (gm/ID )4 , (gm/ID )5 , (gm/ID )6 , (gm/ID )7 , and the dependent parameters are W1 = W2 , W3 = W4 ,
A IDD Av0 + + (2) A0 IDD0 Av Here, A is the silicon area occupied by all transistors, including drain and source regions (estimated), A0 is a reference area for normalization, IDD is the total supply current and Av is the low-frequency gain. The gate transconductance of the input differential pair is set by the GBW and Miller cap, as:
ID1 = “
3
· L3 (8) L 3 The design characteristics calculation is straightforward. The low-frequency gain, for example, is given by W3 =
fc =
gm1 = GBW · Cf
where In1 is the normalized current given by the gm/ID x In curve. The same approach is done for the remaining transistors. For example, the size of the transistors in the current mirror load is:
(5) (6)
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simulated results for the Required ≤ 200 ≥ 60 ≥ 90 ≥ 15 ≥ 15 ≤ −1 ≥1 minimize ≤ 200 minimize
Simulated 102 60 95 15 15 -0.8 1.4 9064 160 1.62
implemented in the LIT tool and provides a reasonable solution in a short CPU time. The main advantage is the simple sizing method based on the transistor inversion coefficient, which is calculated by a single technology-specific characteristic curve gm/ID versus In . The design space is not limited to strong inversion region, but also to moderate and weak inversion, allowing low-power optimum design. Comparing with a typical human-made design procedure, the advantages of using an automated design methodology are the reduced design time, better performance and the possibility of even non-expert analog designers to achieve good solutions for non-critical applications.
5. REFERENCES [1] P. E. Allen and D. R. Holberg. CMOS Analog Circuit Design. Oxford University Press, Oxford, second edition, 2002. [2] R. M. Coitinho, L. H. Spiller, M. C. Schneider, and C. Galup-Montoro. A simplified methodology for the extraction of acm most model parameters. In 14th Symposium on Integrated Circuits and Systems Design, pages 136–141, Bras´ılia, Brazil, September 2001. [3] A. I. A. Cunha, M. C. Schneider, and C. Galup-Montoro. An MOS transistor model for analog circuit design. IEEE Journal of Solid-State Circuits, 33(10):1510–1519, October 1998. [4] B. de Smedt and G. G. E. Gielen. Watson: Design space boundary exploration and model generation for analog and rf ic design. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 22(2):213–224, Feb 2003. [5] M. Degrauwe, O. Nys, E. Dukstra, J. Rijmenants, S. Bitz, B. L. A. G. Goffart, E. A. Vittoz, S. Cserveny, C. Meixenberger, G. V. der Stappen, and H. J. Oguey. IDAC: An interactive design tool for analog CMOS circuits. IEEE Journal of Solid-State Circuits, SC-22(6):1106–1116, December 1987. [6] F. El-Turky and E. E. Perry. BLADES: An artificial intelligence approach to analog circuit design. IEEE Transactions on Computer-Aided Design, 8(6):680–692, June 1989. [7] G. E. Gielen, H. C. C. Walscharts, and W. M. C. Sansen. Analog circuit design optimization based on symbolic simulation and simulated annealing. IEEE Journal of Solid-State Circ., 25(3):707–713, June 1990. [8] G. G. E. Gielen, H. C. C. Walscharts, and W. M. C. Sansen. ISAAC: A symbolic simulator for analog integrated circuits. IEEE Journal of Solid-State Circuits, 24(6):1587–1597, December 1989. [9] A. Girardi and S. Bampi. Lit - an automatic layout generation tool for trapezoidal association of transistors for basic analog building blocks. In Proceedings of the Design Automation and Test in Europe, March 2003. [10] R. Harjani, R. A. Rutenbar, and L. R. Carley. OASYS: A framework for analog circuit synthesis. IEEE Transactions on Computer Aided Design, 8(12), 1989. [11] M. D. M. Hershenson, S. P. Boyd, and T. H. Lee. Optimal design of a CMOS op-amp via geometric programming. IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, 20(1):1–21, January 2001.
Figure 4: Cost function evolution Table 2: Transistors sizes obtained by the optimization method for the Miller amplifier Transistor L(μm) W (μm) gm/ID (V −1 ) M1 , M2 1.2 74.8 16.5 M3 , M4 6.6 15 4.7 0.6 147 15.7 M5 M6 6 10.7 0.7 M7 , M8 0.8 5.5 3.1 W5 , W6 , W7 = W8 , Cf and bias current. The constraints L ≥ Lmin , W ≥ Wmin and (gm/ID )min ≤ (gm/ID ) ≤ (gm/ID )max avoid infeasible solutions, with Lmin = 0.3μm, Wmin = 0.6μm, (gm/ID )min = 0.1 and (gm/ID )max = 25 in our technology. The range of gm/ID is well known from device physics and behaves smoothly over a wide range of transistor biases, which is advantageous for the search robustness. Moreover, the design space is limited by values of gm/ID less than about 28V −1 , which is the theoretical maximum gm/ID of bulk MOS transistors. Design objectives and design specifications are evaluated in terms of free variables (gm/ID )i and Li . The same occurs with the dependent variables such as Wi and IDi . The optimization process for the example took 158 iterations and mere 91 million floating point operations. The final transistors sizes obtained by the iterations with the analytical models are shown in table 2. The third column of table 1 shows the performance of the optimized solution obtained by electrical simulations of the sized circuit with SMASH. Figure 4 shows the evolution of the cost function after each iteration, converging to a minimum value, reaching stability as the cool-down proceeds.
4.
CONCLUSION
The proposed methodology combines the SA algorithm, the gm/ID characteristics and the ACM MOSFET model in the same environment in order to optimize the design of analog circuits based on the transistor inversion coefficient. The main advantage of using the SA over the crude gm/ID technique is that the design space is explored in a more effective way, combining operation in weak and strong inversion to achieve optimum low-power design. The methodology is
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[12] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220(4598):671–680, May 1983.
[13] F. Silveira, D. Flandre, and P. G. A. Jespers. A gm/ID based methodology for the design of CMOS analog circuits and its application to the synthesis of a silicon-on-insulator micropower OTA. IEEE Journal of Solid-State Circuits, 31(9):1314–1319, September 1996.
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